A modal theory $Z$ using the Guaspari witness comparison signs $\leq, <$ is developed. The theory $Z$ is similar to, but weaker than, the theory $R$ of Guaspari and Solovay. Nevertheless, $Z$ proves the independence of the Rosser fixed-point. A Kripke semantics for $Z$ is presented and some arithmetical interpretations of $Z$ are investigated. Then $Z$ is enriched to $ZI$ by adding a new modality sign for interpretability and by axioms expressing some facts about interpretability of theories. Two arithmetical interpretations of $ZI$ are presented. The proofs of the validity of the axioms of $ZI$ in arithmetical interpretations use some strengthening of Solovay's result about interpretability in Godel-Bernays set theory.